Bloch radius, normal families and quasiregular mappings
HTML articles powered by AMS MathViewer
- by Alexandre Eremenko
- Proc. Amer. Math. Soc. 128 (2000), 557-560
- DOI: https://doi.org/10.1090/S0002-9939-99-05141-2
- Published electronically: July 8, 1999
- PDF | Request permission
Abstract:
Bloch’s Theorem is extended to $K$-quasiregular maps $\mathbf {R}^n \to \mathbf {S}^n$, where $\mathbf {S}^n$ is the standard $n$-dimensional sphere. An example shows that Bloch’s constant actually depends on $K$ for $n\geq 3$.References
- Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
- A. Bloch, Ann. Fac. Sci. Toulouse, 17 (1925).
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
- M. Bonk and A. Eremenko, Schlicht regions for entire and meromorphic functions, Preprint, 1998.
- A. V. Chernavskii, Finite-to-one open mappings of manifolds, Mat. Sb., 65 (1964), 357-369, 66 (1964), 471-472. English transl: AMS Transl. (2), 100 (1972).
- P. Gauthier, Covering properties of holomorphic mappings, to be published in Proc. Int. Conf. Several Compl. Var., Postech, June, 1997, AMS Contemp. Math. Series.
- David Minda, Bloch constants for meromorphic functions, Math. Z. 181 (1982), no. 1, 83–92. MR 671716, DOI 10.1007/BF01214983
- Ruth Miniowitz, Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc. 84 (1982), no. 1, 35–43. MR 633273, DOI 10.1090/S0002-9939-1982-0633273-X
- Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 994644, DOI 10.1090/mmono/073
- Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
- Seppo Rickman, The analogue of Picard’s theorem for quasiregular mappings in dimension three, Acta Math. 154 (1985), no. 3-4, 195–242. MR 781587, DOI 10.1007/BF02392472
- G. Valiron, Recherches sur le théorème de M. Picard, Ann. Sci. École Norm. Sup., 38 (1921), 389-430.
- Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817. MR 379852, DOI 10.2307/2319796
- L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc., 35 (1998), 215–230.
Bibliographic Information
- Alexandre Eremenko
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 63860
- Email: eremenko@math.purdue.edu
- Received by editor(s): March 16, 1998
- Received by editor(s) in revised form: April 8, 1998
- Published electronically: July 8, 1999
- Additional Notes: The author was supported by NSF grant DMS-9800084.
- Communicated by: Albert Baernstein II
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 557-560
- MSC (1991): Primary 30C65, 30D45
- DOI: https://doi.org/10.1090/S0002-9939-99-05141-2
- MathSciNet review: 1641689